ABSTRACT
In this chapter we want to investigate the phenomenon that decomposition into irreducible elements need not be unique in a factorization domain R. The mathematician Ernst Eduard Kummer (1810–1893) had the brilliant idea that in such a case uniqueness might be enforced by enlarging the ring R by certain “ideal elements” in a way analogous to the geometric procedure of adjoining “points at infinity” to an affine plane to obtain a projective plane. Vaguely speaking, Kummer’s idea was as follows: If we enlarge the ring, we have more elements available and hence have more possibilities of factoring elements; hence we might be able to further decompose elements which are irreducible in R with the aid of the “ideal elements” adjoined from outside the ring which now might serve as last multiplicative building blocks. This might have the consequence that two essentially different factorizations of a ring element in R can be refined to obtain the same finer factorization into “ideal elements” in both cases. Let us take a typical example for non-unique factorization, say ( 1 + − 5 ) ( 1 − − 5 ) = 2.3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq4290.tif"/> in ℤ [ − 5 ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq4291.tif"/> . Following Kummer, we want to adjoin “ideal elements” p 1, p 2, p 3, p 4 to ℤ [ − 5 ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq4292.tif"/> such that () ( 1 + − 5 ) ︸ = p 1 p 2 ( 1 − − 5 ) ︸ = p 3 p 4 = 2 ︸ = p 1 p 3 . 3 ︸ = p 2 p 4 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq4293.tif"/>
